By exploiting locality, we can perform \(e^{-itV}\)  with \(O\left(n\right)\)  gates. An additional logarithmic factor will be introduced if we implement \(e^{-itT}\)  using the fermionic Fourier transform. However, this can be avoided by further decomposing \(T=T_{\text{odd}}+T_{\text{even}}\)  as in [1]}, so that \(H=T_{\text{odd}}+T_{\text{even}}+V\) . The analysis of Trotter error proceeds in a similar way as in [sec:path]Section sec:path, which establishes the claimed gate complexity of \(\mathcal {O}\left(n\eta ^{1/p}\right)\) .
